THE RESULTS CONCERNING JORDAN DERIVATIONS

Abstract

Let R be a 3!-torsion free semiprime ring, and let D : R ! R be a Jordan derivation on a semiprime ring R: In this case, we show that [D(x); x]D(x) = 0 if and only if D(x)[D(x); x] = 0 for every x 2 R: In particular, let A be a Banach algebra with rad(A): If D is a continuous linear Jordan derivation on A; then we see that [D(x); x]D(x) 2 rad(A) if and only if [D(x); x]D(x) 2 rad(A) for all x 2 A:

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